Theorem csbima12 5191

Description: Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Revised by NM, 20-Aug-2018.)

Assertion

Ref	Expression
csbima12	|- [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B )

Proof of Theorem csbima12

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3352	. . . 4 |- ( y = A -> [_ y / x ]_ ( F " B ) = [_ A / x ]_ ( F " B ) )
2		csbeq1 3352	. . . . 5 |- ( y = A -> [_ y / x ]_ F = [_ A / x ]_ F )
3		csbeq1 3352	. . . . 5 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4	2, 3	imaeq12d 5175	. . . 4 |- ( y = A -> ( [_ y / x ]_ F " [_ y / x ]_ B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
5	1, 4	eqeq12d 2486	. . 3 |- ( y = A -> ( [_ y / x ]_ ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B ) <-> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) ) )
6		vex 3034	. . . 4 |- y e. _V
7		nfcsb1v 3365	. . . . 5 |- F/_ x [_ y / x ]_ F
8		nfcsb1v 3365	. . . . 5 |- F/_ x [_ y / x ]_ B
9	7, 8	nfima 5182	. . . 4 |- F/_ x ( [_ y / x ]_ F " [_ y / x ]_ B )
10		csbeq1a 3358	. . . . 5 |- ( x = y -> F = [_ y / x ]_ F )
11		csbeq1a 3358	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
12	10, 11	imaeq12d 5175	. . . 4 |- ( x = y -> ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B ) )
13	6, 9, 12	csbief 3374	. . 3 |- [_ y / x ]_ ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B )
14	5, 13	vtoclg 3093	. 2 |- ( A e. _V -> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
15		csbprc 3774	. . 3 |- ( -. A e. _V -> [_ A / x ]_ ( F " B ) = (/) )
16		csbprc 3774	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )
17	16	imaeq2d 5174	. . . 4 |- ( -. A e. _V -> ( [_ A / x ]_ F " [_ A / x ]_ B ) = ( [_ A / x ]_ F " (/) ) )
18		ima0 5189	. . . 4 |- ( [_ A / x ]_ F " (/) ) = (/)
19	17, 18	syl6req 2522	. . 3 |- ( -. A e. _V -> (/) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
20	15, 19	eqtrd 2505	. 2 |- ( -. A e. _V -> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
21	14, 20	pm2.61i 169	1 |- [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B )

Syntax hints:   -. wn 3   = wceq 1452   e. wcel 1904   _Vcvv 3031   [_csb 3349   (/)c0 3722   "cima 4842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852

This theorem is referenced by:  csbrn  5304  disjpreima  28271  csbpredg  31797  brtrclfv2  36390  sbcheg  36445